Polytope of Type {6,3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3,6}*1944e
Also Known As : ₆T⁴_(1,1)(3,3). if this polytope has another name.
Group : SmallGroup(1944,2346)
Rank : 4
Schlafli Type : {6,3,6}
Number of vertices, edges, etc : 54, 81, 81, 6
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Locally Toroidal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,3,2}*648, {6,3,6}*648b
   9-fold quotients : {6,3,2}*216, {6,3,6}*216
   27-fold quotients : {2,3,6}*72, {6,3,2}*72
   81-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)
(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)(60,63)
(67,70)(68,71)(69,72)(76,79)(77,80)(78,81);;
s1 := ( 2, 3)( 4, 5)( 7, 9)(10,27)(11,26)(12,25)(13,19)(14,21)(15,20)(16,23)
(17,22)(18,24)(28,55)(29,57)(30,56)(31,59)(32,58)(33,60)(34,63)(35,62)(36,61)
(37,81)(38,80)(39,79)(40,73)(41,75)(42,74)(43,77)(44,76)(45,78)(46,67)(47,69)
(48,68)(49,71)(50,70)(51,72)(52,66)(53,65)(54,64);;
s2 := ( 1,37)( 2,39)( 3,38)( 4,40)( 5,42)( 6,41)( 7,43)( 8,45)( 9,44)(10,28)
(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,47)(20,46)(21,48)
(22,50)(23,49)(24,51)(25,53)(26,52)(27,54)(55,64)(56,66)(57,65)(58,67)(59,69)
(60,68)(61,70)(62,72)(63,71)(73,74)(76,77)(79,80);;
s3 := (28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)
(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)
(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)
(60,63)(67,70)(68,71)(69,72)(76,79)(77,80)(78,81);
s1 := Sym(81)!( 2, 3)( 4, 5)( 7, 9)(10,27)(11,26)(12,25)(13,19)(14,21)(15,20)
(16,23)(17,22)(18,24)(28,55)(29,57)(30,56)(31,59)(32,58)(33,60)(34,63)(35,62)
(36,61)(37,81)(38,80)(39,79)(40,73)(41,75)(42,74)(43,77)(44,76)(45,78)(46,67)
(47,69)(48,68)(49,71)(50,70)(51,72)(52,66)(53,65)(54,64);
s2 := Sym(81)!( 1,37)( 2,39)( 3,38)( 4,40)( 5,42)( 6,41)( 7,43)( 8,45)( 9,44)
(10,28)(11,30)(12,29)(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,47)(20,46)
(21,48)(22,50)(23,49)(24,51)(25,53)(26,52)(27,54)(55,64)(56,66)(57,65)(58,67)
(59,69)(60,68)(61,70)(62,72)(63,71)(73,74)(76,77)(79,80);
s3 := Sym(81)!(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)
(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)
(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);
poly := sub<Sym(81)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;

References :

Theorem 11C7,11C8, McMullen P., Schulte, E.; Abstract Regular Polytopes (\ Cambridge University Press, 2002)

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